Mercurial > hg > Members > atton > delta_monad
annotate agda/delta/monad.agda @ 146:57601209eff3 default tip
Add an example used multi types on Delta
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Tue, 24 Mar 2015 17:04:00 +0900 |
parents | 575de2e38385 |
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rev | line source |
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131 | 1 open import Level |
2 open import Relation.Binary.PropositionalEquality | |
3 open ≡-Reasoning | |
4 | |
88
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5 open import basic |
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6 open import delta |
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7 open import delta.functor |
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8 open import nat |
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9 open import laws |
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10 |
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11 module delta.monad where |
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12 |
108
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13 |
107
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14 tailDelta-to-tail-nt : {l : Level} {A B : Set l} (n m : Nat) |
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15 (f : A -> B) (d : Delta (Delta A (S (S m))) (S n)) -> |
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16 delta-fmap tailDelta (delta-fmap (delta-fmap f) d) ≡ delta-fmap (delta-fmap f) (delta-fmap tailDelta d) |
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17 tailDelta-to-tail-nt O O f (mono (delta x d)) = refl |
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18 tailDelta-to-tail-nt O (S m) f (mono (delta x d)) = refl |
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19 tailDelta-to-tail-nt (S n) O f (delta (delta x (mono xx)) d) = begin |
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20 delta (mono (f xx)) |
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21 (delta-fmap tailDelta (delta-fmap (delta-fmap f) d)) |
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22 ≡⟨ cong (\de -> delta (mono (f xx)) de) (tailDelta-to-tail-nt n O f d) ⟩ |
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23 delta (mono (f xx)) |
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24 (delta-fmap (delta-fmap f) (delta-fmap tailDelta d)) |
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25 ∎ |
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26 tailDelta-to-tail-nt (S n) (S m) f (delta (delta x (delta xx d)) ds) = begin |
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27 delta (delta (f xx) (delta-fmap f d)) |
108
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28 (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds)) |
107
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29 ≡⟨ cong (\de -> delta (delta (f xx) (delta-fmap f d)) de) (tailDelta-to-tail-nt n (S m) f ds) ⟩ |
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30 delta (delta (f xx) (delta-fmap f d)) |
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31 (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds)) |
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32 ∎ |
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33 |
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34 |
105
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35 delta-eta-is-nt : {l : Level} {A B : Set l} -> {n : Nat} |
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36 (f : A -> B) -> (x : A) -> (delta-eta {n = n} ∙ f) x ≡ delta-fmap f (delta-eta x) |
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37 delta-eta-is-nt {n = O} f x = refl |
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38 delta-eta-is-nt {n = S n} f x = begin |
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39 (delta-eta ∙ f) x ≡⟨ refl ⟩ |
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40 delta-eta (f x) ≡⟨ refl ⟩ |
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41 delta (f x) (delta-eta (f x)) ≡⟨ cong (\de -> delta (f x) de) (delta-eta-is-nt f x) ⟩ |
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42 delta (f x) (delta-fmap f (delta-eta x)) ≡⟨ refl ⟩ |
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43 delta-fmap f (delta x (delta-eta x)) ≡⟨ refl ⟩ |
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44 delta-fmap f (delta-eta x) ∎ |
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45 |
107
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46 |
105
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47 delta-mu-is-nt : {l : Level} {A B : Set l} {n : Nat} -> (f : A -> B) -> (d : Delta (Delta A (S n)) (S n)) |
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48 -> delta-mu (delta-fmap (delta-fmap f) d) ≡ delta-fmap f (delta-mu d) |
107
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49 delta-mu-is-nt {n = O} f (mono d) = refl |
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50 delta-mu-is-nt {n = S n} f (delta (delta x d) ds) = begin |
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51 delta (f x) (delta-mu (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds))) |
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52 ≡⟨ cong (\de -> delta (f x) (delta-mu de)) (tailDelta-to-tail-nt n n f ds ) ⟩ |
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53 delta (f x) (delta-mu (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds))) |
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54 ≡⟨ cong (\de -> delta (f x) de) (delta-mu-is-nt f (delta-fmap tailDelta ds)) ⟩ |
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55 delta (f x) (delta-fmap f (delta-mu (delta-fmap tailDelta ds))) |
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56 ∎ |
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57 |
105
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58 |
107
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59 delta-fmap-mu-to-tail : {l : Level} {A : Set l} (n m : Nat) (d : Delta (Delta (Delta A (S (S m))) (S (S m))) (S n)) -> |
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60 delta-fmap tailDelta (delta-fmap delta-mu d) |
105
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61 ≡ |
107
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62 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta d))) |
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63 delta-fmap-mu-to-tail O O (mono (delta d ds)) = refl |
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64 delta-fmap-mu-to-tail O (S n) (mono (delta (delta x (delta xx d)) (delta (delta dx (delta dxx dd)) ds))) = refl |
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65 delta-fmap-mu-to-tail (S n) O (delta (delta (delta x (mono xx)) (mono (delta dx (mono dxx)))) ds) = begin |
108
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66 delta (mono dxx) (delta-fmap tailDelta (delta-fmap delta-mu ds)) |
107
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67 ≡⟨ cong (\de -> delta (mono dxx) de) (delta-fmap-mu-to-tail n O ds) ⟩ |
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68 delta (mono dxx) |
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69 (delta-fmap delta-mu |
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70 (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) |
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71 ∎ |
137 | 72 delta-fmap-mu-to-tail (S n) (S m) (delta (delta (delta x (delta xx d)) (delta (delta dx (delta dxx dd)) ds)) dds) = begin |
107
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73 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) |
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74 (delta-fmap tailDelta (delta-fmap delta-mu dds)) |
137 | 75 ≡⟨ cong (\de -> delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) de) (delta-fmap-mu-to-tail n (S m) dds) ⟩ |
107
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76 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) |
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77 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta dds))) |
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78 ∎ |
105
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79 |
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80 |
88
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81 |
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82 -- Monad-laws (Category) |
131 | 83 -- association-law : join . delta-fmap join = join . join |
84 delta-association-law : {l : Level} {A : Set l} {n : Nat} (d : Delta (Delta (Delta A (S n)) (S n)) (S n)) -> | |
105
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85 ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) |
137 | 86 delta-association-law {n = O} (mono d) = refl |
131 | 87 delta-association-law {n = S n} (delta (delta (delta x d) dd) ds) = begin |
108
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88 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds))) |
107
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89 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-fmap-mu-to-tail n n ds) ⟩ |
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90 delta x (delta-mu (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) |
131 | 91 ≡⟨ cong (\de -> delta x de) (delta-association-law (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) ⟩ |
107
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92 delta x (delta-mu (delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) |
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93 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-mu-is-nt tailDelta (delta-fmap tailDelta ds) ) ⟩ |
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94 delta x (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds)))) |
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95 ∎ |
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96 |
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97 |
144
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98 delta-left-unity-law : {l : Level} {A : Set l} {n : Nat} (d : Delta A (S n)) -> (delta-mu ∙ delta-eta) d ≡ id d |
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99 delta-left-unity-law (mono x) = refl |
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100 delta-left-unity-law (delta x d) = begin |
105
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|
101 (delta-mu ∙ delta-eta) (delta x d) |
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102 ≡⟨ refl ⟩ |
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103 delta-mu (delta-eta (delta x d)) |
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104 ≡⟨ refl ⟩ |
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105 delta-mu (delta (delta x d) (delta-eta (delta x d))) |
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106 ≡⟨ refl ⟩ |
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107 delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) |
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108 ≡⟨ refl ⟩ |
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109 delta x (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) |
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110 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-eta-is-nt tailDelta (delta x d))) ⟩ |
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111 delta x (delta-mu (delta-eta (tailDelta (delta x d)))) |
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112 ≡⟨ refl ⟩ |
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113 delta x (delta-mu (delta-eta d)) |
144
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114 ≡⟨ cong (\de -> delta x de) (delta-left-unity-law d) ⟩ |
105
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115 delta x d |
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116 ≡⟨ refl ⟩ |
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117 id (delta x d) ∎ |
88
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118 |
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119 |
144
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120 delta-right-unity-law : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> |
105
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121 (delta-mu ∙ (delta-fmap delta-eta)) d ≡ id d |
144
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122 delta-right-unity-law (mono x) = refl |
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123 delta-right-unity-law {n = (S n)} (delta x d) = begin |
105
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124 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩ |
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125 delta-mu ( delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩ |
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126 delta-mu (delta (delta-eta x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ |
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127 delta (headDelta {n = S n} (delta-eta x)) (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) ≡⟨ refl ⟩ |
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128 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) |
131 | 129 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-covariant tailDelta delta-eta d)) ⟩ |
105
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130 delta x (delta-mu (delta-fmap (tailDelta ∙ delta-eta {n = S n}) d)) ≡⟨ refl ⟩ |
144
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131 delta x (delta-mu (delta-fmap (delta-eta {n = n}) d)) ≡⟨ cong (\de -> delta x de) (delta-right-unity-law d) ⟩ |
105
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132 delta x d ≡⟨ refl ⟩ |
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133 id (delta x d) ∎ |
88
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134 |
105
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135 |
104
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|
136 |
105
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137 delta-is-monad : {l : Level} {n : Nat} -> Monad {l} (\A -> Delta A (S n)) delta-is-functor |
94
bcd4fe52a504
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90
diff
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|
138 delta-is-monad = record { eta = delta-eta; |
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|
139 mu = delta-mu; |
105
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140 eta-is-nt = delta-eta-is-nt; |
115
e6bcc7467335
Temporary commit : Proving association-law ...
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108
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|
141 mu-is-nt = delta-mu-is-nt; |
131 | 142 association-law = delta-association-law; |
144
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143 right-unity-law = delta-right-unity-law ; |
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144 left-unity-law = \x -> (sym (delta-left-unity-law x)) } |
104
ebd0d6e2772c
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|
145 |
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|
146 |
88
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147 |
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148 |
96
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|
149 |
88
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150 {- |
96
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|
151 |
88
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152 -- Monad-laws (Haskell) |
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153 -- monad-law-h-1 : return a >>= k = k a |
105
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154 monad-law-h-1 : {l : Level} {A B : Set l} -> |
88
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155 (a : A) -> (k : A -> (Delta B)) -> |
96
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|
156 (delta-return a >>= k) ≡ (k a) |
88
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|
157 monad-law-h-1 a k = refl |
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|
158 |
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|
159 |
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|
160 |
526186c4f298
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|
161 -- monad-law-h-2 : m >>= return = m |
96
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|
162 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= delta-return) ≡ m |
88
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|
163 monad-law-h-2 (mono x) = refl |
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|
164 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
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|
165 |
526186c4f298
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|
166 |
526186c4f298
Split monad-proofs into delta.monad
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|
167 |
96
dfe8c67390bd
Unify Levels in delta
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|
168 -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g |
105
e6499a50ccbd
Retrying prove monad-laws for delta
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|
169 monad-law-h-3 : {l : Level} {A B C : Set l} -> |
96
dfe8c67390bd
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|
170 (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) -> |
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|
171 (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g) |
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|
172 monad-law-h-3 (mono x) f g = refl |
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|
173 monad-law-h-3 (delta x d) f g = begin |
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94
diff
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|
174 (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩ |
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parents:
94
diff
changeset
|
175 (delta-bind (delta-bind (delta x d) f) g) ∎ |
88
526186c4f298
Split monad-proofs into delta.monad
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diff
changeset
|
176 |
96
dfe8c67390bd
Unify Levels in delta
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94
diff
changeset
|
177 |
dfe8c67390bd
Unify Levels in delta
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94
diff
changeset
|
178 |
dfe8c67390bd
Unify Levels in delta
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94
diff
changeset
|
179 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
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parents:
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diff
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|
180 -} |